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Mirrors > Home > MPE Home > Th. List > reupick3 | Structured version Visualization version GIF version |
Description: Restricted uniqueness "picks" a member of a subclass. (Contributed by Mario Carneiro, 19-Nov-2016.) |
Ref | Expression |
---|---|
reupick3 | ⊢ ((∃!𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ∧ 𝑥 ∈ 𝐴) → (𝜑 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-reu 3111 | . . . 4 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
2 | df-rex 3110 | . . . . 5 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝜓))) | |
3 | anass 469 | . . . . . 6 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ 𝜓) ↔ (𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝜓))) | |
4 | 3 | exbii 1830 | . . . . 5 ⊢ (∃𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ 𝜓) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝜓))) |
5 | 2, 4 | bitr4i 279 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ ∃𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ 𝜓)) |
6 | eupick 2687 | . . . 4 ⊢ ((∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∧ ∃𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ 𝜓)) → ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓)) | |
7 | 1, 5, 6 | syl2anb 597 | . . 3 ⊢ ((∃!𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓)) → ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓)) |
8 | 7 | expd 416 | . 2 ⊢ ((∃!𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓)) → (𝑥 ∈ 𝐴 → (𝜑 → 𝜓))) |
9 | 8 | 3impia 1110 | 1 ⊢ ((∃!𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ∧ 𝑥 ∈ 𝐴) → (𝜑 → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1080 ∃wex 1762 ∈ wcel 2080 ∃!weu 2610 ∃wrex 3105 ∃!wreu 3106 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1778 ax-4 1792 ax-5 1889 ax-6 1948 ax-7 1993 ax-10 2111 ax-12 2140 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-ex 1763 df-nf 1767 df-sb 2042 df-mo 2575 df-eu 2611 df-rex 3110 df-reu 3111 |
This theorem is referenced by: reupick2 4211 fvineqsneq 34237 |
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